You are given an integer n. Your goal is to reduce it to $0$ by repeatedly performing either of the following bit operations:

• Flip the rightmost bit (bit at position $0$) of n.

• Flip the bit at position $i$ (for $i > 0$) only if the bit at position $i - 1$ is $1$ and all bits from position $i - 2$ down to $0$ are set to $0$.

Determine and return the minimum number of these operations required to reduce n to $0$.

Constraints:

• $0 \leq$n $\leq 10^9$

First, we need to analyze how binary numbers can be manipulated using the two allowed operations:

• Operation 1: Flip the rightmost (0th) bit at any time.

• Operation 2: Flip the bit at position $i$ (for $i > 0$) only if the bit at position $i - 1$ is $1$ and all bits from position $i - 2$ down to $0$ are set to $0$.

Now, to understand the core of this problem, let’s dive into the logical analysis of this problem to reach the solution:

Analyze numbers with exactly one bit set (powers of 2)

Let’s start with the simplest case numbers with exactly one bit set (powers of two) such as$2 (10), 4 (100), 8 (1000)$, etc.

• $1 (1)$ takes exactly $1$ operation: $1 → 0$

• $2 (10)$ takes exactly $3$ operations: $10 → 11 → 01 → 00$

• $4 (100)$ takes exactly $7$ operations: $100 → 101 → 111 → 110 → 010 → 011 → 001 → 000$ ...